Isolation of regular graphs and $k$-chromatic graphs

For any graph $G$ and any set $\mathcal{F}$ of graphs, let $\iota(G,\mathcal{F})$ denote the size of a smallest set $D$ of vertices of $G$ such that the graph obtained from $G$ by deleting the closed neighbourhood of $D$ does not contain a copy of a graph in $\mathcal{F}$. Thus, $\iota(G,\{K_1\})$ is the domination number of $G$. For any integer $k \geq 1$, let $\mathcal{F}_{0,k} = \{K_{1,k}\}$, let $\mathcal{F}_{1,k}$ be the set of regular graphs of degree at least $k-1$, let $\mathcal{F}_{2,k}$ be the set of graphs whose chromatic number is at least $k$, and let $\mathcal{F}_{3,k}$ be the union of $\mathcal{F}_{0,k}$, $\mathcal{F}_{1,k}$ and $\mathcal{F}_{2,k}$. We prove that if $G$ is a connected $n$-vertex graph and $\mathcal{F} = \mathcal{F}_{0,k} \cup \mathcal{F}_{1,k}$, then $\iota(G, \mathcal{F}) \leq \frac{n}{k+1}$ unless $G$ is a $k$-clique or $k = 2$ and $G$ is a $5$-cycle. This generalizes a bound of Caro and Hansberg on the $\{K_{1,k}\}$-isolation number, a bound of the author on the cycle isolation number, and a bound of Fenech, Kaemawichanurat and the author on the $k$-clique isolation number. By Brooks' Theorem, the same holds if $\mathcal{F} = \mathcal{F}_{3,k}$. The bounds are sharp.